arXiv:0911.3354v2 [math.NT] 18 Nov 2009
APPROXIMATE GROUPS AND THEIR APPLICATIONS: WORK
OF BOURGAIN, GAMBURD, HELFGOTT AND SARNAK
BEN GREEN
Abstract. This is a survey of several exciting recent results in which techniques originating in the area known as additive combinatorics have been applied to give results in other areas, such as group theory, number theory and
theoretical computer science. We begin with a discussion of the notion of an
approximate group and also that of an approximate field, describing key results of Freı̆man-Ruzsa, Bourgain-Katz-Tao, Helfgott and others in which the
structure of such objects is elucidated. We then move on to the applications.
In particular we will look at the work of Bourgain and Gamburd on expansion
properties of Cayley graphs on SL2 (Fp ) and at its application in the work of
Bourgain, Gamburd and Sarnak on nonlinear sieving problems.
1. Introduction
The subject of additive combinatorics has grown enormously over the last ten
years and now comprises a large collection of tools with many applications in number theory and elsewhere, for example in group theory and theoretical computer
science. It has often been thought a little difficult to specify to an outsider exactly
what the subject is 1. However the following point of view seems to be gradually crystallising: additive combinatorics is the study of approximate mathematical
structures such as approximate groups, rings, fields, polynomials and homomorphisms. It is interested in what the right definitions of these approximate structures
are, what can be said about them, and what applications this has to other parts of
mathematics.
This article has three main aims. Firstly, we wish to introduce the above point
of view to a general audience, focussing in particular on the basic theory of approximate groups and approximate fields. Secondly, we wish to sketch some beautiful
applications of these ideas. One of them has to do with the beautiful picture on the
cover (for which we thank Cliff Reiter) of an Apollonian circle packing. It is classical
that the radii of these circles are all reciprocals of integers. We will describe work
of Bourgain, Gamburd and Sarnak giving upper bounds for the number of circles
at “depth n” which have radius the reciprocal of a prime. Thirdly, we wish to hint
at the extraordinary variety of different areas of mathematics which have started
to interact with additive combinatorics: geometric group theory, analytic number
theory, model theory and point-set topology are just the ones we shall mention
here.
2000 Mathematics Subject Classification. Primary .
This article was written while the author was a fellow at the Radcliffe Institute at Harvard. It
is a pleasure to thank the institute for its support and excellent working conditions.
1
See, for example, my own attempt in the opening remarks of [26].
1
2
BEN GREEN
What we offer here is merely a taste of this viewpoint of additive combinatorics
as the theory of approximate structure and of its applications. We do not touch
on the theory of approximate polynomials (a.k.a. the theory of Gowers norms)
or say much at all about approximate homomorphisms, or anything about the
many applications of these two notions. These topics will be covered in detail in
forthcoming lecture notes of the author [29].
2. Approximate groups
Before we can define an approximate group, we need to recall what an exact one
is. We shall be concerned with finite groups, and we shall be working inside some
ambient group G, so that it makes sense to talk about multiplication of elements
and taking inverses. If A ⊆ G is a finite set then we shall write A · A := {a1 a2 :
a1 , a2 ∈ A} and A · A−1 := {a1 a−1
2 : a1 , a2 ∈ A}. Later on we shall see more general
notations such as A · A · A and A · B whose meaning, we hope, will be evident. The
following proposition, whose proof is an exercise in undergraduate group theory,
gives various criteria for A to be a subgroup or something very closely related.
Proposition 2.1. Let A be a finite subset of some ambient group G. Then we have
the following statements2:
(1) |A · A−1 | > |A|, with equality if and only if A = Hx for some subgroup
H 6 G and some element x ∈ G;
(2) |A · A| > |A|, with equality if and only if A = Hx for some subgroup H 6 G
and some element x in the normaliser NG (H);
(3) The number of quadruples (a1 , a2 , a3 , a4 ) ∈ A4 with a1 a−1
= a3 a−1
is at
2
4
3
most |A| , with equality if and only if A = Hx for some subgroup H 6 G
and some element x ∈ G;
(4) The number of quadruples (a1 , a2 , a3 , a4 ) ∈ A4 with a1 a2 = a3 a4 is at most
|A|3 , with equality if and only if A = Hx for some subgroup H 6 G and
some element x ∈ NG (H);
(5) P(a1 a2 ∈ A|a1 , a2 ∈ A) 6 1, with equality if and only if A = H for some
subgroup H 6 G;
(6) P(a1 a−1
2 ∈ A|a1 , a2 ∈ A) 6 1, with equality if and only if A = H for some
subgroup H 6 G.
This would be a rather odd proposition to see formulated in an algebra text.
However each of the statements (1) – (6) has been constructed as an inequality in
such a way that one may ask when equality approximately holds. Before we can
talk about such approximate variants, however, we need to know how approximate
they will be. For this purpose we introduce a parameter K > 1; larger values of K
will indicate more approximate, and thus less structured, objects3.
2Statements (5) and (6) look “probabilistic” but this is just a notation. By P(a a ∈ A|a , a ∈
1 2
1 2
A) we mean simply the proportion of all pairs a1 , a2 ∈ A for which a1 a2 also lies in A.
3In practice the theory when K ≈ 1 is very different from the theory when, for example,
K ∼ 100. In the former setting, these approximate notions of subgroup constitute very small
perturbations of the exact characterisations of Proposition 2.1, and it turns out (though is not
always trivial to prove) that the approximate objects so defined are small perturbations of the
exact objects characterised by Proposition 2.1. In conversation Tao and I tend to refer to this
regime as “the 99% world”, an expression I would not be averse to popularising. In this paper
K will be much larger, causing the theory to become much richer. Tao and I call this the “1%
APPROXIMATE GROUPS AND THEIR APPLICATIONS
3
Consider, then, the following list of properties that a finite set A ⊆ G might
enjoy.
(1) |A · A−1 | 6 K|A|;
(2) |A · A| 6 K|A|;
−1
(3) The number of quadruples (a1 , a2 , a3 , a4 ) ∈ A4 with a1 a−1
is at
2 = a3 a4
3
least |A| /K;
(4) The number of quadruples (a1 , a2 , a3 , a4 ) ∈ A4 with a1 a2 = a3 a4 is at least
|A|3 /K;
(5) P(a1 a2 ∈ A|a1 , a2 ∈ A) > 1/K;
(6) P(a1 a−1
2 ∈ A|a1 , a2 ∈ A) > 1/K.
Now these are by no means as closely equivalent as the properties (1) – (6) in
Proposition 2.1. Let us give an example in which the ambient group is Z, and
where we use additive rather than multiplicative notation. Take A = {1, . . . , n} ∪
{2n+1 , 2n+2 , . . . , 22n }. Then it is easy to check that (3) and (4) are both satisfied
with any K > 12, as n → ∞, this being because there are 32 n3 (1 + o(1)) solutions
to a1 + a2 = a3 + a4 with a1 , a2 , a3 , a4 ∈ {1, . . . , n}. On the other hand the sumset
A + A contains the numbers 2n+i + j for each pair i, j with 0 < i, j 6 n. Since
these numbers are all distinct, we have |A + A| > n2 = |A|2 /2, which means that if
n is sufficiently large depending on K then (2) is not satisfied at all.
Rather remarkably, however, there is a sense in which the concepts (1) – (6) are
all roughly the same. To say what we mean by that, we introduce the following
notion of rough equivalence4.
Definition 2.2 (Rough Equivalence). Suppose that A and B are two finite sets
in some ambient group and that K > 1 is a parameter. Then we write A ∼K
B to mean that there is some x in the ambient group such that |A ∩ Bx| >
max(|A|, |B|)/K. We say that A and B are roughly equivalent (with parameter
K).
The remarkable fact alluded to above is the following. For every choice of j, j 0 ∈
{1, . . . , 6}, suppose that some set A satisfies condition (j) in the list above with
parameter K. Then there is a set B satisfying condition (j 0 ) with parameter K 0 =
poly(K) (some polynomial in K) such that A ∼K 0 B. Of particular note is the
fact that the weak “statistical” properties (3) – (6) imply the apparently more
structured properties (1) and (2). The proof of this is not at all trivial and the main
content of it is the so-called Balog-Szemerédi-Gowers theorem [22], generalised to
the nonabelian setting in the fundamental paper of Tao [59], as well as a collection
of “sumset estimates” which, in the abelian case, I refer to collectively as Ruzsa
calculus [28]. These estimates of Ruzsa have such a classical role in the theory that
we record two of them, in the abelian setting, here: we will mention these two again
later on.
world” although the parameter K could be anything between 2 (say) and some small power of
|A|.
4The fact that we have written Bx rather than xB is a little arbitrary. The notion of rough
equivalence will, in this survey, be applied to classes of sets (such as (1) – (6) here) which are
invariant under conjugation, in which case whether we multiply on the left or on the right in the
definition makes little difference.
4
BEN GREEN
Theorem 2.3 (Ruzsa). Suppose that A1 , A2 and A3 are finite sets in some ambient
abelian group. Then |A1 ||A2 − A3 | 6 |A1 − A2 ||A1 − A3 | and |A1 + A1 | 6 |A1 −
A1 |3 /|A1 |2 .
The original paper [47], the book [63] or the notes [28] may be consulted for
more details. The first estimate is true in general groups but adapting the second
requires care: see [59].
It might be remarked that for many pairs (j) and (j 0 ) the correspondence between
the relevant properties is a little tighter than mere rough equivalence, and often
this can be useful. We shall not dwell on this point here. In the paper of Tao just
mentioned one finds what has become the “standard” notion of an approximate
group.
Definition 2.4 (Approximate group). Suppose that A is a finite subset of some
ambient group and that K > 1 is a parameter. Then we say that A is a Kapproximate group if it is symmetric (that is, if a ∈ A then a−1 ∈ A, and the
identity lies in A) and if there is a set X in the ambient group with |X| 6 K and
such that A · A ⊆ X · A.
This notion, it turns out, is roughly equivalent to (1) - (6) above. It has certain
advantages over (1) - (6), for example as regards its behaviour under homomorphisms. It is also clear that an approximate group in this sense enjoys good control
of iterated sumsets. Thus, for example, A · A · A ⊆ X · X · A, which means that
|A3 | = |A · A · A| 6 K 2 |A|, and similarly |An | 6 K n−1 |A| where An denotes the set
of all products a1 . . . an with a1 , . . . , an ∈ A. From now on, when we speak of an
approximate group, we will be referring primarily to Definition 2.4.
With this discussion in mind, we can introduce what might be termed the rough
classification problem of approximate group theory.
Question 2.5. Consider the collection C of all K-approximate groups A in some
ambient group G. Is there some “highly structured” subcollection C 0 such that
every A ∈ C is roughly equivalent to some set B ∈ C 0 with parameter K 0 , where K 0
depends only on K?
This question has been addressed in a great many different contexts, starting
with the Freı̆man-Ruzsa theorem [20, 48], which gives an answer for subsets of
Z. Here, it is possible to take C 0 to consist of the so-called generalised arithmetic
progressions, that is to say sets B of the form
B := {l1 x1 + · · · + ld xd : li ∈ Z, |li | 6 Li },
where x1 , . . . , xd ∈ R, the quantities L1 , . . . , Ld are “lengths” and d 6 K. Note in
particular that, even in the highly abelian setting of the integers Z, approximate
groups are a more general kind of object than genuine subgroups. That is, the
theory of approximate groups, even up to rough equivalence, is a little richer than
the theory of finite subgroups of Z (which is in fact a rather trivial theory). The
remarkable feature of the Freı̆man-Ruzsa theorem is that the theory is not much
richer, in the sense that generalised progressions remain highly “algebraic” objects.
Here is a list of other contexts in which the question has been at least partially
answered:
• abelian groups [30];
APPROXIMATE GROUPS AND THEIR APPLICATIONS
5
• nilpotent and solvable groups [10, 11, 19, 50, 61];
• free groups [46];
• linear groups SL2 (R) [16],
SL2 (C) [13, 33, 34],
SL3 (Z) [13],
SL3 (C) (sketched in [34]),
“bounded” subsets of SLn (C) including Un (C) [12],
SL2 (Fp ) [33],
SL3 (Fp ) [34]
and SL2 (Z/qZ) for various other q (cf. [4]).
It is generally felt that approximate groups in quite general contexts can be
controlled by objects built up from genuine subgroups and nilpotent objects; this
has been found in all of the examples just mentioned and is suggested by the
famous theorem of Gromov on groups with polynomial growth [32] and the recent
quantitative formulation of it due to Shalom and Tao [55]. Quite precise suggestions
along these lines have been made by Helfgott, Lindenstrauss and others: more
information on this can be found on Tao’s blog [62].
Before leaving this subject, we remark that even (perhaps especially) in the
abelian case the issue of the dependence of K 0 on K is far from being resolved. No
examples are known to rule out the possibility that, with the right definition of the
highly-structured class C 0 , K 0 can be taken to be polynomial in K. In particular
this is conjectured when the ambient group G is FZ2 , the countable infinite vector
space over the field of two elements, and C 0 consists of (finite) subgroups of G.
This assertion5 is known as the polynomial Freı̆man-Ruzsa conjecture [49], see also
[25, 27]. It is equivalent to the following question which, for many years, I have
tried to advertise to those for whom the word cohomology holds no fear.
Question 2.6. Suppose that φ : Fn2 → FZ2 is a map such that φ(x + y) − φ(x) − φ(y)
takes on at most K different values as x, y range over Fn2 . Is it true that φ = φ̃ + η,
where φ̃ is linear and η takes on at most K 0 = poly(K) different values?
It is a very easy exercise to obtain such a statement with K 0 = 2K but, so far
as I know, no serious improvement of this bound has ever been obtained6.
3. Approximate rings and fields
Fortified by the experiences of the last section, one might attempt to come up
with a sensible notion of an approximate ring. A natural one, based perhaps on (2)
in the previous section, is as follows: if A is a finite subset of some ambient ring R,
we say that it is a K-approximate ring if |A + A| 6 K|A| and |A · A| 6 K|A|. Here,
of course, A + A := {a1 + a2 : a1 , a2 ∈ A} and A · A = {a1 a2 : a1 , a2 ∈ A} as before.
If R = F is actually a field (or an integral domain, which embeds into its field
of fractions) then we refer to A as an approximate field, noting that approximate
closure under division is essentially automatic in view of the rough equivalence of
the notions (1) and (2) of approximate group.
5There are variants of this conjecture over other groups, such as Z; see [23, 31].
6I would be very interested to see even a bound of the form 2o(K) .
6
BEN GREEN
The study of approximate rings and fields was initiated in a paper of Erdős and
Szemerédi [17] who proved (though not in this language!) that a K-approximate
subfield of Z must have size poly(K). They in fact conjectured that the right
bound is C K 1+ for any > 0, but this is so far unresolved; the best exponent so
far obtained is 3 + , a result of Solymosi [57]. Note that this is equivalent to, and
more usually stated as, the lower bound
max(|A + A|, |A · A|) > c |A|4/3+
for all finite sets A ⊆ Z. In a different paper [56], Solymosi generalised the ErdősSzemerédi result to show that every K-approximate subfield of C has size at most
212 K 4 .
The general theory of approximate fields can be said to have started with the
papers of Bourgain-Katz-Tao [8] and Bourgain-Glibichuk-Konyagin [7, 9], where7
the following result is established.
Theorem 3.1. Let p be a prime and let K > 2. Then every K-approximate subfield
of Fp has size at most K C or at least K −C p, for some absolute constant C.
The arguments on page 384 of [7], though they are phrased in a more limited
context, essentially prove that every approximate subfield (in an arbitrary ambient
field) must be roughly equivalent to a genuine finite subfield. This unifies the results
of Erdős-Szemerédi and Solymosi with Theorem 3.1. In fact something similar is
true for approximate rings, at least provided the ambient ring R does not have “too
many” zero divisors. These issues are comprehensively explored in an interesting
paper [60] of Tao, which also has a very comprehensive collection of references.
Suppose that A is a K-approximate field in some ambient field F, that is to
say both |A · A| and |A + A| are bounded by K|A|. We are going to sketch a
proof that F must contain a genuine subfield B which is “close” to A. The first
step is to prove the Katz-Tao lemma, which asserts that A (or, more precisely, a
large subset A0 ⊆ A) behaves in a manner which more strongly resembles that
of a field: that is to say, A is almost closed under both addition/subtraction and
multiplication/division simultaneously. To give a (relevant) example, the set
a1 − a3
A := {a5 + a6
: a1 , . . . , a6 ∈ A}
a4 − a2
has size K|A|, where K = poly(K).
A slick proof of the Katz-Tao lemma is given in [60, Section 2.5] and we shall
say little more about it here other than to remark that it involves a combination
of Ruzsa’s sumset calculus and clever elementary arguments. Personally, I regard
it as part of the “basic” theory of approximate fields as opposed to the “structural
theory”, to be regarded on the same level as the arguments used to show that definitions (1) – (6) of an approximate group are roughly equivalent (namely, Ruzsa’s
sumset calculus and the Balog-Szemerédi-Gowers theorem). In other words one
might argue that the smallness of A, or of similar objects, might be taken as an
alternative definition of approximate field.
7The original paper [8] of Bourgain, Katz and Tao did not quite classify the very small (smaller
than pδ ) approximate subrings of Fp ; this restriction was removed in [7, 9]. Very often the
approximate fields under consideration in a given setting will have size at least pδ , and for this
reason one often refers to the Bourgain-Katz-Tao theorem.
APPROXIMATE GROUPS AND THEIR APPLICATIONS
7
Suppose that A is known to have this property, that is to say |A| 6 K|A|. Then
it is possible to establish an intriguing dichotomy: if ξ ∈ F× then either
(3.1)
|A + Aξ| = |A|2
or
(3.2)
|A + Aξ| 6 K|A|.
Here, A + Aξ refers to the set of all a1 + a2 ξ with a1 , a2 ∈ A. To see why this is
so, note that |A + Aξ| 6 |A|2 and that equality occurs if and only if the elements
a1 + a2 ξ are all distinct. If equality does not occur then we may find a nontrivial
3
solution to a1 + a2 ξ = a3 + a4 ξ, which means that ξ = aa14 −a
−a2 . But then every
element of A + Aξ has the form
a5 + a6 ξ = a5 + a6
a1 − a3
,
a4 − a2
and thus lies in A.
On the other hand, it is not hard to see using Ruzsa calculus8 that if ξ1 , ξ2 satisfy
(3.2) then for ξ = ξ1 + ξ2 , ξ1 − ξ2 , ξ1 ξ2 , ξ1 ξ2−1 we have
C
0
|A + ξ · A| 6 K |A| 6 K C |A|
for absolute constants C, C 0 . If K is a sufficiently small power of |A| then this
means that (3.1) cannot hold, forcing us to conclude that (3.2) holds for ξ. In
this way we identify the set9 of all ξ satisfying (3.2) as a genuine subfield of F.
Straightforward additional arguments allow one to show that this subfield and F
are roughly equivalent.
The original argument of [8] is different and specific to Fp but rather fun and,
given the preceding discussion, it is not hard to say a few meaningful words about it.
Suppose for the sake of illustration that A ⊆ Fp is a K-approximate subfield of size
∼ p1/10 ; our task is to derive a contradiction if (say) K = po(1) . Suppose that the
Katz-Tao lemma has already been applied, so that A, as defined above, is known to
be small. The sets A, A, . . . arising from (boundedly many) successive applications
of this operation may also be shown to be small. Now simple averaging arguments
(using nothing more than the fact that |A| = p1/10 ) show that Fp has dimension
at most 100 (say) as a “vector space” over A; that is, there exist x1 , . . . , x100 ∈ Fp
such that
(3.3)
Fp = Ax1 + · · · + Ax100 .
Now x1 , . . . , x100 cannot be a “basis” for Fp over A since otherwise we would have
p = |A|100 , contrary to the assumption that p is prime. Thus there must exist some
x ∈ Fp which is representable in two different ways as
x = a1 x1 + · · · + a100 x100 = a01 x1 + · · · + a0100 x100
8In addition to the bounds of Theorem 2.3 one requires an inequality controlling |A +A +A |
1
2
3
in terms of the |Ai + Aj |.
9Note that this set may be identified with A−A .
(A−A)×
8
BEN GREEN
with a1 , . . . , a100 , a01 , . . . , a0100 ∈ A. Suppose, without loss of generality, that a100 6=
a0100 . Then
(a1 − a01 )x1 + · · · + (a99 − a099 )x99
x100 =
.
a0100 − a100
By substituting this expression for x100 into (3.3), we see that
Fp = Ax1 + · · · + Ax99 .
Repeating the argument gives (without loss of generality)
Fp = Ax1 + · · · + Ax98 ,
and we may continue in this fashion to get, eventually,
˙˙
Fp = Ax1 .
This is contrary to the fact that none of the sets A, A, . . . has size much larger than
that of A itself, namely about p1/10 , and a contradiction ensues.
Remarkably, the main “dimension reduction” idea here comes from a paper in
point-set topology, namely Edgar and Miller’s solution of the Erdős-Volkmann ring
problem [15] (that is, the statement that all Borel subrings of R have dimension 0
or 1). See in particular Lemma 1.3 of that paper.
4. Helfgott’s results
In this section we discuss the results of Helfgott [33, 34] concerning approximate
subgroups of
a b
SL2 (Fp ) := {
: a, b, c, d ∈ Fp : ad − bc = 1}.
c d
Helfgott proves the following.
Theorem 4.1 (Helfgott). Suppose that A ⊆ SL2 (Fp ) is a K-approximate group.
Then A is roughly K C -equivalent to an upper-triangular K C -approximate subgroup
of SL2 (Fp ) (that is, an approximate subgroup conjugate to a set of upper-triangular
matrices).
Rather than discuss Helfgott’s result itself, we discuss the analogous question for
SL2 (C). Here the answer is rather simpler and is given in [13], based on Helfgott’s
work.
Theorem 4.2. Suppose that A ⊆ SL2 (C) is a K-approximate group. Then A is
roughly K C -equivalent to an abelian K C -approximate subgroup of SL2 (C).
If desired the abelian approximate group could itself be controlled by a generalised progression using the Freı̆man-Ruzsa theorem.
We will only sketch a proof of the weaker result that A is K C -equivalent to an
upper-triangular K C -approximate subgroup, that is to say the direct analogue of
Helfgott’s result. In SL2 (C), additional arguments may then be applied to prove
APPROXIMATE GROUPS AND THEIR APPLICATIONS
9
Theorem 4.2; there are no such arguments in SL2 (Fp ), since the upper-triangular
“Borel subgroup”
λ
µ
{
: λ ∈ F∗p , µ ∈ Fp }
0 λ−1
is not close to abelian.
The proof of this weak form of Theorem 4.2 is simpler than that of Theorem
4.1 in two major ways. Firstly since C is algebraically closed we may talk about
eigenvalues, eigenvectors and diagonalization without the need to pass to an extension field, whereas over Fp we would have to involve the quadratic extension Fp2 .
Secondly, the structure of K-approximate subfields of C is easy to describe: by the
theorem of Solymosi [56] they are all sets of size at most 212 K 4 . Theorem 3.1, by
contrast, has to allow for those approximate fields which are almost all of Fp . Worse
still, to handle SL2 (Fp ) Helfgott must in fact classify approximate subfields of Fp2 ,
and this involves the additional possibility of sets which are close to the subfield
Fp .
For the sake of exposition, we will assume in the first instance that A is a
genuine finite subgroup of SL2 (C); our task is to show that A contains a large
upper-triangular subgroup. When we have sketched how Helfgott’s argument looks
in this case we will remark on the additional technicalities required to make the
argument “robust” enough to apply to K-approximate groups.
The key idea in Helfgott’s argument, referred to by subsequent authors as trace
amplification, involves examining the set of traces
tr A := {tr a : a ∈ A}.
We will sketch a proof that a large subset of this set of traces is a 224 -approximate
subfield of C of size greater than 2108 . This contradicts Solymosi’s theorem [56]
and so we must be in one of those degenerate situations. Careful analysis of each
of them leads to the conclusion that A is roughly upper-triangular.
The first degenerate situation to analyse is that in which tr A is small, an appropriate notion of small being | tr A| 6 2111 . Now a linear algebra computation
(Lemma 4.2 of [6]) confirms that if g, h ∈ A are elements without a common eigenvector in C2 then the map
SL2 (C) → C3 : x 7→ (tr x, tr(gx), tr(hx))
is at most two-to-one. This, or rather the fact that something like this holds, is
not at all surprising: indeed knowledge of tr(x), tr(gx), tr(hx) together with the
fact that det(x) = 1 provides four pieces of information which, generically, ought
to more-or-less determine the four entries of the matrix x. If A contains two such
elements g, h then it follows that we have
|A| 6 2| tr A|3 6 2334 ,
and so |A| is also small10. If, by contrast, A does not contain two such elements, and
if |A| > 3, then it is easy to see that there is some v ∈ C2 which is an eigenvector
for all of A simultaneously. With respect to a basis containing v, every matrix in
A is upper-triangular.
10Additive combinatorics has a bad reputation for referring to quantities like 2334 as “small”.
“Bounded by an absolute constant” might be more appropriate.
10
BEN GREEN
Suppose, then, that | tr A| > 2111 . In particular (!) there is some element g ∈ A
which is non-parabolic, or in other words tr g 6= ±2; such elements have distinct
eigenvalues and so are diagonalisable.
Write A0 ⊆ A for the set of non-parabolic elements; then | tr A0 | > | tr A| −
2 > 12 | tr A|. Now in SL2 (C) the trace of a non-parabolic element g completely
determines the conjugacy class of g. It follows that there is some non-parabolic
g ∈ A such that the conjugacy class of A containing g has size at most 2|A|/| tr A|.
By the orbit-stabiliser theorem, the centraliser11
T = CA (g) = {a ∈ A : ag = ga}
1
2 | tr A|.
But by changing basis so that g is in diagonal form (with
has size at least
distinct diagonal entries) it is not hard to check that T consists entirely of diagonal
matrices. No single trace can arise from more than two of these elements, and so
| tr T | > 14 | tr A| > 2109 . We shall show that the set
R := {tr a2 : a ∈ T }
is a 224 -approximate subfield of C. Noting that
|R| > 21 | tr T | > 2108 ,
(4.1)
this is contrary to Solymosi’s theorem. In order to do this we play around a little
with traces. Such playing around is most productive if, in the basis just selected,
a11 a12
there is an element a =
∈ A with a11 a12 a21 a22 6= 0. The absence of
a21 a22
such an element is another degenerate situation to analyse, and once again one can
check12 that A must be either upper-triangular or else equal to one of the dihedral
groups, each of which has an index two abelian subgroup.
Now let us note that
R · R ⊆ R + R,
(4.2)
this being a consequence of the fact that
−2
−2 −2
−2 2
2
2 2
2 −2
(t21 + t−2
1 )(t2 + t2 ) = (t1 t2 + t1 t2 ) + (t1 t2 + t1 t2 ).
Let us also note that
−1
t1 t2
0
t1 t2
tr
−1 a
0
t−1
t
0
1 2
0
t−1
1 t2
−2
2
a−1 = µ(t21 + t−2
1 ) + λ(t2 + t2 ),
where µ := a11 a22 6= 0 and λ := −a12 a21 6= 0, which means that
λR + µR ⊆ tr A.
In particular
µ
R| = |λR + µR| 6 | tr A| 6 16|R|,
λ
which, by Ruzsa’s inequalities (Theorem 2.3, applied with A1 = µλ R and A2 = A3 =
−R) implies that |R + R| 6 224 |R|. This, together with (4.2), implies that R is a
224 -approximate subring of C. By Solymosi’s theorem this implies that |R| 6 2108 ,
contrary to (4.1).
In the above sketch we assumed, of course, that A was actually a finite subgroup.
However the argument was of a type that can be made to work for K-approximate
|R +
11T is for torus, the word used for such a subgroup in Lie theory.
12This is, admittedly, a somewhat tedious check.
APPROXIMATE GROUPS AND THEIR APPLICATIONS
11
groups also. To explain what we mean by this let us remark, rather vaguely, on
how one or two of the steps adapt and then offer some general remarks.
Orbit-Stabiliser theorem. If A is a group and if x ∈ A then we used the fact
that the size of the conjugacy class Σ(x) containing x and that of the centraliser
CA (x) are related by |Σ(x)||CA (x)| = |A|. In fact we only used the inequality
|CA (x)| > |A|/|Σ(x)|, giving us an element with large centraliser, and here is a
simple way of seeing why this holds: all of the conjugates axa−1 , a ∈ A, lie in Σ(x),
and so by the pigeonhole principle there must be distinct elements a1 , . . . , ak ∈ A,
−1
k > |A|/|Σ(x)|, with a1 xa−1
= · · · = ak xa−1
1
k . But then the elements ai a1 ,
i = 1, . . . , k, centralise x. Now if A is only a K-approximate group then this
argument does not quite work, as there is no well-defined notion of conjugacy class.
However a similar pigeonhole argument nonetheless gives us an element with large
centraliser, since the conjugates axa−1 are all constrained to lie in A3 , a set of size
at most K 2 |A|.
Escape from subvarieties. A more interesting point concerns the location of an
element of A which, in a given basis, has no zero entries. Whilst this might not be
a priori possible if A is only an approximate group, it is possible to find such an
element in An for some bounded n (independent of the approximation parameter
K), and this is good enough for Helfgott’s purposes. This is a special case of a nice
lemma of Eskin, Mozes and Oh [18] called “escape from subvarieties”. The point
is that the group hAi generated by A, if it is not almost upper-triangular, contains
an element with no zero entries – indeed this fact was used in the above sketch. In
other words, hAi is not contained in the subvariety of SL2 (C) defined by
x11 x12
V := {
: x11 x12 x21 x22 = 0}.
x21 x22
The Eskin-Mozes-Oh result states that in such a situation we can find “evidence” for
the non-containment of hAi inside V by taking just a bounded number, depending
only on V , of products of A.
It seems, then, that certain types of argument – in some sense those involving
“bounded length” computations in the ambient group – adapt very well from the
traditional group theory setting to approximate groups. At the moment we do
not have anything approaching a precise formulation of this principle and indeed
at present the passage from the “exact” to the approximate is as much an art as
a science. Nonetheless, there seems to be merit in looking for “bounded length”
proofs in traditional group theory which might be adapted to the approximate
setting. Perhaps this is as good a place as any to mention the remarkable recent
paper of Hrushovski [36] in which tools from model theory have been applied to
the study of approximate groups. The ramifications of that paper are not yet
completely clear, but it looks as though Theorem 1.3 of that paper together with
some structure theory of algebraic groups ought to lead, without too much difficulty,
to a proof of the following statement.
Conjecture 4.3. Suppose that A ⊆ SLn (C) is a K-approximate group. Then
there is a K 0 -approximate group B which is nilpotent and K 0 -controls A, where K 0
depends only on K.
It seems reasonable to conjecture that K 0 can be taken to depend polynomially
on K, although in their present form Hrushovski’s techniques will not give this.
12
BEN GREEN
5. Cayley graphs on SL2 (Fp )
We move on now to applications of the theory of approximate groups. In this
section we discuss the paper [3] of Bourgain and Gamburd. This paper concerns
expander graphs. For the purposes of this discussion these are 2k-regular graphs
Γ on n vertices for which there is a constant c > 0 such that for any set X of at
most n/2 vertices of Γ, the number of vertices outside X which are adjacent to X
is at least c|X|. Expander graphs share many of the properties of random regular
graphs, and this is an important reason why they are of great interest in theoretical
computer science. There are many excellent articles on expander graphs ranging
from the very concise [51] to the seriously comprehensive [35].
A key issue is that of constructing explicit expander graphs, and in particular
that of constructing families of expanders in which k and c are fixed but the number
n of vertices tends to infinity. Many constructions have been given, and several
of them arise from Cayley graphs. Let G be a finite group and suppose that
S = {g1±1 , . . . , gk±1 } is a symmetric set of generators for G. The Cayley graph
C(G, S) is the 2k-regular graph on vertex set G in which vertices x and y are joined
if and only if xy −1 ∈ S. Such graphs provided some of the earliest examples of
expanders [41, 42]. A natural way to obtain a family of such graphs is to take some
large “mother” group G̃ admitting many homomorphisms π from G̃ to finite groups,
a set S̃ ⊆ G̃, and then to consider the family of Cayley graphs C(π(G̃), π(S̃)) as π
ranges over a family of homomorphisms. The work under discussion concerns the
case G̃ = SL2 (Z), which of course admits homomorphisms πp : SL2 (Z) → SL2 (Fp )
for each prime p. For certain sets S̃ ⊆ G̃, for example
±1 ±1
1 1
1 0
S̃ = {
,
}
0 1
1 1
or
1
S̃ = {
0
2
1
±1 1
,
2
0
1
±1
},
spectral methods from the theory of automorphic forms may be used to show that
(C(πp (G̃), πp (S̃)))p prime is a family of expanders. See [40] and the references therein.
These methods depend on the fact that the group hS̃i has finite index in G̃ = SL2 (Z)
and they fail when this is not the case, for example when
±1 ±1
1 3
1 0
(5.1)
S̃ = {
,
}.
0 1
3 1
In [40] Lubotzky asked whether the corresponding Cayley graphs in this and other
cases might nonetheless form a family of expanders, the particular case of (5.1)
being known as his “1-2-3 question”. The paper of Bourgain and Gamburd under
discussion answers this quite comprehensively, showing that all that is required is
that the group generated by S̃ is not virtually abelian (contains a finite index abelian
subgroup). We will sketch the proof in the case that S̃ generates a nonabelian free
subgroup of SL2 (Z). This is essentially the most general case, since the kernel of
the natural homomorpism from hS̃i to SL2 (F2 ) ∼
= Sym(3) is free and has index at
most 6 in hS̃i.
APPROXIMATE GROUPS AND THEIR APPLICATIONS
13
Theorem 5.1 (Bourgain – Gamburd). Let G̃ = SL2 (Z) as above and suppose that
S̃ is a finite symmetric set generating a free subgroup of SL2 (Z). Then
(C(πp (G̃), πp (S̃)))p prime
is a family of expanders.
The notation we have introduced here is rather cumbersome, so let us write
Γp := C(πp (G̃), πp (S̃)). For concreteness
we will
focus
on the special case S̃ =
1
3
1
0
{A, A−1 , B, B −1 }, where A =
and B =
are the matrices relevant
0 1
3 1
to Lubotzky’s 1-2-3 question. The argument is almost identical in any other case.
In this case, then, Γp is the graph on vertex set SL2 (Fp ) in which x is joined to y
if and only if xy −1 is one of the elements A, A−1 , B or B −1 considered modulo p.
Supposing that p > 3, each of these graphs is 4-regular. The number of verices in
Γp is n := | SL2 (Fp )| = p(p2 − 1).
The reader may be interested to see a proof, using the “ping-pong” technique of
Felix Klein, that that the subgroup of SL2 (Z) generated by these A and B is indeed
free. Consider the natural action of A and B on the projective plane P1 (Q). Write
X := {(λ : 1) ∈ P1 (Q) : |λ| < 1}
and
Y := {(1 : λ) ∈ P1 (Q) : |λ| < 1},
and observe that X and Y are disjoint and jouent au ping pong, that is to say
An (X) ⊆ Y
for all n ∈ Z \ {0}
and
B n (Y ) ⊆ X
for all n ∈ Z \ {0}.
(The origin of the name should be clear – the “players” A and B hit the domains
X and Y back and forth – as should the preference for the French term rather
than the cumbersome “play table tennis with one another”.) If the group generated by A and B is not free, then some nontrivial reduced word in A and B
is equal to the identity, where “reduced word” means a finite word of the form
. . . Am1 B n1 . . . Amk B nk . . . with m1 , n1 , . . . , mk , nk 6= 0. The conjugate of such a
word by an appropriate power of A will still be the identity and will now have the
form w = Am1 B n1 . . . Amk B nk Amk+1 with mi , nj 6= 0. However by repeated application of the ping-pong properties we see that w(X) ⊆ Y , certainly an impossibility
since X and Y are disjoint and w is supposed to be the identity.
Following that slight digression let us focus once again on the Cayley graphs Γp ,
our aim being to prove that they form a family of expanders as p ranges over the
primes. To do this we begin by giving a spectral interpretation of the expansion
property which we defined combinatorially above. For each p we may consider the
Laplacian of the corresponding Cayley graph, that is to say the operator
∆ : L2 (SL2 (Fp )) → L2 (SL2 (Fp ))
defined by
∆f (x) := f (x) − 41 (f (Ax) + f (A−1 x) + f (Bx) + f (B −1 x)).
The eigenvalues of the Laplacian lie in the interval [0, 2]. Zero is certainly an
eigenvalue, since ∆1 = 0. Write the eigenvalues in ascending order as 0 = λ0 6
14
BEN GREEN
λ1 6 . . . 6 λn−1 . It turns out the expansion properties of the graph Γp (in fact of
any regular graph) are intimately connected with the size of the second-smallest
eigenvalue λ1 = λ1 (Γp ). The precise relation between the combinatorial property
of expansion and this spectral property is discussed in Section 2 of [35], but for
our purposes we need only remark that it suffices to show that the second-smallest
eigenvalue λ1 (Γp ) is bounded away from zero independently of n (in fact, this is
also a necessary condition for expansion). The term spectral gap is used to describe
this property: there is a gap at the bottom of the spectrum in which there are no
eigenvalues apart from zero.
To try to show that there is a spectral gap, consider the operator
T : L2 (SL2 (Fp )) → L2 (SL2 (Fp ))
given by T := 4(id −∆), that is to say
T f (x) := f (Ax) + f (A−1 x) + f (Bx) + f (B −1 x).
The matrix13 of T is same thing as the adjacency matrix of the graph Γp , that is
to say the matrix whose xy entry is 1 if x ∼ y and zero otherwise. The eigenvalues
of T are of course µi = 4(1 − λi ), i = 0, . . . , n − 1, and it is a very well-known and
Pn−1
easy to establish fact that the 2mth moment i=0 µ2m
is equal to n times W2m ,
i
the number of closed walks of length 2m from the identity to itself. It follows that
we have
!
n−1
X
1 2m
2m
(1 − λi )
.
1+
(5.2)
W2m = 4
n
i=1
Note in particular that W2m > n1 42m , since all the terms are non-negative. At first
glance it looks as though the only way to use (5.2) to bound λ1 away from zero
would be to get rather precise estimates on W2m , and in particular one would at the
very least want to show that W2m < n2 42m . However a remarkable observation, used
earlier in related contexts by Sarnak and Xue [54] and Gamburd [21], comes into
play. This is that any eigenspace of the Laplacian is SL2 (Fp )-invariant, where the
action of SL2 (Fp ) on L2 (SL2 (Fp )) is the right-regular one given by g◦f (x) := f (xg).
In other words, any such eigenspace has the structure of a representation of SL2 (Fp )
and thus, by basic representation theory, decomposes into irreducible representation
of SL2 (Fp ). But by a classical theorem of Frobenius all such representations have
dimension at least (p − 1)/2 ∼ n1/3 . This means that λ1 = λ2 = · · · = λl for some
l ∼ n1/3 , and hence from (5.2) we in fact have the bound
1
(5.3)
W2m 2/3 42m (1 − λ1 )2m .
n
This enables a meaningful spectral gap (lower bound on λ1 ) to be obtained from
somewhat weaker upper bounds on W2m .
The main new content of [3], then, is to obtain those upper bounds on W2m , the
number of walks of length 2m starting and finishing at the identity, for appropriate
m. A nice way of thinking about these walks is in terms of convolution powers of
the probability measure
ν := 41 (δA + δA−1 + δB + δB −1 )
13With respect to the basis of SL (F ) consisting of the functions 1 : SL (F ) → C defined
t
2 p
2 p
by 1t (x) = 1 if x = t and 0 otherwise.
APPROXIMATE GROUPS AND THEIR APPLICATIONS
15
on SL2 (Fp ), where δg (x) = n if x = g and 0 otherwise. This measure ν is a very
singular or “spiky” probability measure, supported on just the four points A, A−1 , B
and B −1 . Now the convolution
ν (2) := ν ∗ ν(x) := Ey∈SL2 (Fp ) ν(xy −1 )ν(y)
is supported on words of length at most two in A, A−1 , B and B −1 , or alternatively
those x in the graph Γp which can be reached from the identity by a path of length
two, the value of ν ∗ ν(x) being 4−2 n times the number of paths of length two from
the identity to x. Similarly higher convolution powers ν (m) (x) := ν ∗ · · · ∗ ν(x) give
4−m n times the number of paths of length m from the identity to x. The idea
of the proof is to examine these convolution powers, showing that they become
progressively more “spread out” until, for suitable m, ν (2m) vaguely resembles the
uniform measure 1 which assigns weight one to each point of SL2 (Fp ). Then, in
particular, ν (2m) (0) ∼ 1, meaning that W2m ∼ 42m /n. Combined with (5.3), this
is enough to establish the desired spectral gap.
The notion of a probability measure µ on SL2 (Fp ) being “spread out” may be
quantified using the L2 -norm
1/2
kµk2 := Ex∈SL2 (Fp ) µ(x)2
.
The L2 -norm of a delta measure δg is n1/2 , which is huge, whilst that of the uniform measure 1 is equal to one, the smallest value possible by the Cauchy-Schwarz
inequality. It is not hard to show that convolution cannot increase the L2 -norm,
and so we have the chain of inequalities
(5.4)
n1/2 = kν (1) k2 > kν (2) k2 > . . . .
The aim is to show that this sequence is, in fact, rather rapidly decreasing. Roughly
speaking one shows that
(5.5)
kν (m1 ) k2 ≈ 1
for some m1 ≈ C1 log p; this m1 turns out to be an appropriate choice to substitute
into (5.3) in order to reach the desired conclusions.
It turns out that this sequence gets off to a rather good start. This is a consequence of an observation of Margulis [43], namely that the freeness of the subgroup of SL2 (Z) generated by A and B persists to some extent even when reduced modulo p. Indeed let us take a reduced word w = Am1 B n1 . . . Amk B nk with
m1 , . . . , mk , n1 , . . . , nk 6= 0 and suppose that this equals the identity when reduced
modulo p, that is to say in SL2 (Fp ). Lifting back up to SL2 (Z) we have
w̃ = Am1 B n1 . . . Amk B nk ≡ id
(mod p).
But the freeness of the lifted group means that w̃ 6= id, and thus in order to be
congruent to the identity mod p the matrix w̃ must have at least one entry of size
at least p − 1. But by some simple matrix inequalities this is impossible provided
that
|m1 | + |n1 | + · · · + |mk | + |nk | < c log p
for some absolute constant c > 0.
It follows that the subgroup of SL2 (Fp ) generated by A and B is “free up to
words of length c log p”. In terms of the Cayley graphs Γp this means that up to
retracing steps there is a unique walk of length m between the identity and x for
16
BEN GREEN
any x ∈ SL2 (Fp ) and for any m 6 m0 := c log p/2. This implies that the measures
ν (m) , m > m0 are already rather spread out. To quantify this (and in particular
to deal with the issue of “retracing steps”) a result of Kesten concerning random
walks in the free group may be applied. The conclusion is that
(5.6)
kν (m0 ) k2 n1/2−γ
for some γ > 0. This is good progress on the way to (5.5) and represents a significant
improvement on the initial bound kν (1) k2 = n1/2 .
It is convenient to imagine, for the rest of the argument, that all probability
n
measures µ on G have the form µ(x) = |A|
1A (x) for some set A ⊆ G, the “support”
of µ. Whilst this is clearly not true, various (somewhat technical) decompositions
into level sets may be used to reduce to this case. For such a measure we have
kµk2 = (n/|A|)1/2 ,
and so the bound (5.6) corresponds to |A| n2γ , certainly a reasonable level of
spreadoutness.
The rest of the argument, which constitutes the heart of the paper, involves
examining the convolution powers between ν (m0 ) and ν (m1 ) for a suitable m1 ∼
C1 log p, the aim being to establish (5.5). An application of the “dyadic pigeonholing argument”, used to great effect by Bourgain in many papers, is employed: if
kν (m1 ) k2 is much larger than 1, this means that the sequence (5.4) cannot decay too
rapidly between ν (m0 ) and ν (m1 ) and so there must be two convolution powers ν (m)
and ν (2m) , m0 6 m < m1 , such that kν (2m) k2 ≈ kν (m) k2 . Let us be deliberately
vague about the meaning of ≈ here.
n
1A (x) for some set A ⊆ G. Noting that ν (2m) =
Suppose that ν (m) (x) = |A|
ν (m) ∗ ν (m) , it is not hard to compute that the ratio
kν (2m) k22 /kν (m) k22
is actually equal to |A|−3 times the number of quadruples a1 , a2 , a3 , a4 ∈ A4 with
a1 a2 = a3 a4 . This may be compared with condition (4) in the list of properties
which are known to roughly characterise approximate groups. Thus, being even
rougher at this point,
1
1H
H
for some approximate group H ⊆ SL2 (Fp ). Note that the rough equivalence of
(4) and other, more flexible definitions such as Definition 2.4 is one of the deeper
equivalences mentioned in §2, being reliant on the nonabelian Balog-SzemerédiGowers theorem of Tao [59].
(5.7)
ν (m) ∼
If H is already all of SL2 (Fp ) then (5.7) is telling us that ν (m) is close to the
uniform distribution, in which case so is ν (m1 ) , hence (5.5) is established and we
are done. If not then we apply Helfgott’s result, Theorem 4.1, to conclude that
H is essentially upper-triangular, and hence that ν (m0 ) has significant mass on an
upper-triangular subgroup of SL2 (Fp ).
The support of ν (m0 ) , however, consists of words of length at most m0 in the
generators A, A−1 , B and B −1 and, as we stated, these elements behave freely up
to words of this length. This is highly incompatible with upper-triangularity, which
APPROXIMATE GROUPS AND THEIR APPLICATIONS
17
in particular implies that we always have the commutator relation14
(5.8)
[[g1 , g2 ], [g3 , g4 ]] = id .
A pleasant group-theoretic argument formalises this incompatibility and allows one
to show that any set of words of length at most m0 in the generators A, A−1 , B
and B −1 satisfying (5.8) has size at most m60 . This represents a tiny proportion of
the set of all such words, which (counted with multiplicity at least) has cardinality
4m0 . This contradiction finishes the sketch proof of Theorem 5.1.
Before moving on, we wish to record, for use in the next section, a further
observation concerning the measures ν (m) . We sketched a proof that kν (m1 ) k2 ≈
1 for some m1 ∼ C1 log p, that is to say ν (m1 ) vaguely resembles the uniform
distribution on SL2 (Fp ). By taking further convolutions and using the fact that
irreducible representations have large degree once more, this may be bootstrapped
to show that ν (m) becomes exponentially well uniformly-distributed:
(5.9)
ν (m) (x) = 1 + O(ne−cm )
for some absolute c > 0 and for all m. Alternatively, such a statement can be
deduced directly from the spectral gap property, as is done for example in [6, §3.3].
It is interesting to ask whether the arguments might adapt to deal with Cayley
graphs on SLn (Fp ) with n > 3. A recent paper of Bourgain and Gamburd [5] shows
that this is the case when n = 3. The argument is, in large part, quite similar to
the above, except of course that Helfgott’s theorem on approximate subgroups of
SL2 (Fp ) must be replaced by his more difficult result [34] on approximate subgroups
of SL3 (Fp ). There is one significant extra difficulty, however, which is that there are
proper subgroups of SL3 (Fp ) which are not close to upper-triangular, an obvious
example being a copy of SL2 (Fp ). To deal with this a deep algebro-geometric
result of Nori [45] is brought into play, which states that any proper subgroup of
SL3 (Fp ), p sufficiently large, must satisfy a non-trivial polynomial equation. To
obtain a contradiction, it must be shown that the set of words of length m0 in the
generators A and B (say) does not concentrate on the corresponding subvariety
of SL3 (C), and here techniques from the theory of random matrix products and a
certain amount of “quantitative algebraic geometry” are brought into play.
6. Nonlinear sieving problems
In this section we discuss work of Bourgain, Gamburd and Sarnak [6]. The goal
of sieve theory, traditionally viewed as a part of analytic number theory, is to find
prime numbers or at least to say something about them. Historically, the sieve
arose through work of Brun and Merlin on the twin prime problem, that is to say
the problem of finding infinitely may primes p such that p + 2 is also prime. Whilst
this remains a famous open problem, approximations to it have been found. For
example, Brun established the following result.
Theorem 6.1 (Brun). There are infinitely many integers n such that n(n + 2) has
at most 9 prime factors.
14In other words, upper-triangular subgroups of SL (F ) are 2-step solvable.
2 p
18
BEN GREEN
Much later, Chen [14] replaced 9 by 3. One way of stating this type of result is
as follows: there are infinitely many n for which both n(n + 2) is a 3-almost prime,
that is to say a positive integer with at most 3 prime factors.
The aim of [6] is to discover almost primes in more exotic locales, and specifically
in orbits of linear groups. We will sketch a proof of the following result.
Theorem 6.2 (Bourgain-Gamburd-Sarnak). Let A and B be two matrices in
SL2 (Z) generating a free subgroup. Then there is some r such that this group
contains infinitely many r-almost prime matrices (matrices, the product of whose
entries is r-almost prime).
Henceforth we shall say “almost prime” instead of “r-almost prime for some
r”. We
case we focussed on in the last section, when
remark
that in the specific
1 3
1 0
A=
and B =
, the theorem as stated follows from classical sieve
0 1
3 1
theory
of the typeused to prove Brun’s theorem. Indeed (for example) An BA =
9n + 1 30n + 3
, and the product of the entries here is 2·32 ·5·(9n+1)·(10n+1),
3
10
which will be almost prime for infinitely many n by a simple variant of Brun’s
analysis. The issue here is that the subgroup generated by A and B contains
unipotent elements (in this case both A and B are themselves unipotent).
We start with a (very) elementary discussion of what a sieve is. Suppose one has
a finite set X of integers and that one wishes to find primes or almost primes in X.
The most naı̈ve way to do this would be to try to adapt the sieve of Eratosthenes,
using the inclusion-exclusion principle to compute
#{primes in X} = |X|−|X2 |−|X3 |−|X5 |−· · ·+|X6 |+|X10 |+|X15 |+· · ·−|X30 |−. . .
where Xq is the set of elements of X which are divisible by q. Unfortunately it
is well-known that, even when X is an extremely simple set such as {1, . . . , n}, it
is not generally possible to evaluate |Xq | sufficiently accurately to avoid the error
terms in this long sum blowing up. In this simple case just mentioned, for example,
we have |Xq | = bn/qc. However the floor function is rather unpleasant and it is
tempting to write instead |Xq | = n/q + O(1), but then one finds that there are so
many O(1) errors that the sieve of Eratosthenes becomes useless.
By and large, sieve theory is concerned with what it is possible to say about
primes or almost primes in X given “reasonably nice” information about the size
of the sets Xq . Although the sieve of Eratosthenes is bad, other sieves fare rather
better. These other sieves are generally cleverly weighted versions of the sieve of
Eratosthenes, but we will not dwell upon their construction here. A typical example
of “reasonably nice” information about |Xq | would be
|Xq | = β(q)|X| + rq
γ
for all squarefree q 6 |X| , where β(q) is some pleasant multiplicative function and
the error rq is small in the sense that |rq | |X|1−γ for some γ > 0. For example,
if X = {1, . . . , n} then this is true with β(q) = 1/q and for any γ 6 1.
The fundamental theorem of the combinatorial sieve states, roughly speaking,
that such information is enough to find almost primes in X; in fact, one can even
estimate the number of almost primes. What is meant by “almost prime” – that
is, how many prime factors these numbers will have – depends on how large we can
APPROXIMATE GROUPS AND THEIR APPLICATIONS
19
take γ as well as on the so-called dimension of the sieve, which has to do with the
average size of the quantities β. We will not delay ourselves by expanding upon
the details here. Let us instead refer the reader to [6] for the precise formulation
convenient to the application there and to the book [38] or the unpublished notes
[37] for a more wide-ranging discussion of sieves in general with full proofs.
All we shall take from the preceding discussion is the notion that, given a finite
set X to be sieved in order to locate almost primes, we should be looking for good
asymptotics for the size of the sets |Xq |, q squarefree. Returning to Theorem 6.2,
the first obvious question to answer is that of what the set X to be sieved should
be. The set in which we wish to find almost primes is
x1 x2
A := {x1 x2 x3 x4 :
∈ hA, Bi}.
x3 x4
Now A is of course an infinite set of integers. Rather than truncate in the usual
way and take X = A ∩ {1, . . . , N }, it is much more natural to truncate in a manner
that respects the group structure more. This we do by taking
x1 x2
X := {x1 x2 x3 x4 :
∈ Σm (A, B)},
x3 x4
where
Σm (A, B) = {U1 U2 . . . Um : Ui ∈ {A, A−1 , B, B −1 }}
is the set of words of length m in A, A−1 , B and B −1 and X is counted with
multiplicity so that |X| = 4m .
Suppose that p is a prime. Then |Xp | is equal to the number of words w ∈
Σm (A, B), counted with multiplicity, which, when reduced modulo p, give rise to a
matrix in SL2 (Fp ) with at least one zero entry. Writing S ⊆ SL2 (Fp ) for the set of
such matrices, it is easy to compute that |S| = 2(2p − 1)(p − 1). Now the number
of words w ∈ Σm (A, B) which reduce modulo p to some x ∈ SL2 (Fp ) is, in the
notation of the last section, precisely n1 |X|ν (m) (x), and so
X
1
|Xp | = |X|
ν (m) (x).
n
x∈S
However at the end of the last section we saw15 that ν (m) (x) becomes very close
to 1. In fact, in (5.9) we noted the bound ν (m) (x) = 1 + O(ne−cm ). Using this we
obtain
|Xp | = β(p)|X| + rp
where β(p) := 2(2p − 1)/p(p + 1) and |rp | = |X|1−γ for some γ > 0.
Thus the expansion property of the Cayley graphs (C(πp (G̃), πp (S̃)))p prime gives
exactly the kind of information that can be input into the combinatorial sieve!
There is, however, a very major caveat. What we have just said applies only to
Xp when p is a prime, and for the sieve one must understand Xq when q is a general squarefree number. To do this requires the establishment of Theorem 5.1 for
the family (C(πq (G̃), πq (S̃)))q , where now q ranges over all squarefrees and not just
over primes. The broad scheme of the proof is the same, but every single ingredient
must be generalised to the more general setting, starting from the classification of
15Either as a byproduct of the proof, or a consequence, of the expansion property of the family
of Cayley graphs Γp = C(πp (G), πp (S̃)).
20
BEN GREEN
Figure 1. Apollonian circle packing
approximate subrings of Z/qZ. The situation here is more complicated because this
ring will in general have many approximate subrings, namely Z/q 0 Z with q 0 |q. One
of the main technical results of [6] (occupying some 20 pages) is the statement that,
very roughly speaking, these are the only approximate subrings of Z/qZ. Although
this is a deeply technical argument of a type that this author would struggle to
summarise meaningfully even to an expert audience, it might be compared with
the 92-page proof [2] of the corresponding assertion without the squarefree assumption on q. Thankfully16 this is not required for the present application. Once the
classification of approximate subrings of Z/qZ for q squarefree is in place a suitable analogue of Helfgott’s argument is applied to roughly classify approximate
subgroups of SL2 (Z/qZ). Even the statement of this result (Proposition 4.3 in the
paper) is rather technical. Finally, the majority of the argument outlined in the
last section in the case q prime goes over without substantial change.
This concludes our discussion of the proof of Theorem 6.2. To conclude this
survey, we wish to mention a beautiful application, mentioned in the original paper
[6] and in other articles such as [52], of these nonlinear sieving ideas. This has to
do with Apollonian packings such as the one in the attractive image above.
16This is one of the most extraordinarily long and technical arguments the author has ever seen.
The theory of approximate rings when there are many zero-divisors seems to be very difficult.
APPROXIMATE GROUPS AND THEIR APPLICATIONS
21
For a very pleasant and gentle introduction to Apollonian packings, see [1].
Referring to Figure 1, inside each circle is an integer which represents the curvature
of that circle, or in other words the reciprocal of the radius. Some of the number
theory associated with the integers that arise in this way is discussed in the letter
[53] where, for example, it is shown that infinitely many of these curvatures are
prime and in fact that there are infinitely many touching pairs of circles with prime
curvature.
Now a pleasant exercise in Euclidean geometry gives a theorem of Descartes,
namely that the relation between the four integers a1 , a2 , a3 , a4 inside four mutually
touching circles is given by
(6.1)
2(a21 + a22 + a23 + a24 ) = (a1 + a2 + a3 + a4 )2 .
Examples of quadruples (a1 , a2 , a3 , a4 ) which are related in this way and easily
visible in the picture are (13, 21, 24, 124) and (13, 24, 37, 156).
Take a quadruple (C1 , C2 , C3 , C4 ) of touching circles with curvatures
(a1 , a2 , a3 , a4 ) = (13, 21, 24, 124).
There is another circle C10 tangent to C2 , C3 and C4 , and it has curvature a01 = 325.
To find a general relation between a1 and a01 we may note that a1 , a01 are roots of
(6.1) regarded as a quadratic in a1 and thereby obtain the relation
a01 = −a1 + 2a2 + 2a3 + 2a4 .
This may of course be written as
 0 
a1
−1
a2   0
 =
a3   0
a4
0
 
a1
2 2 2
 
1 0 0
 a2  .
0 1 0 a3 
0 0 1
a4
That is, if one starts with some fixed vector such as x0 = (13, 21, 24, 124) then one
may obtain another quadruple of curvatures of circles in the Apollonian packing by
applying the matrix


−1 2 2 2
 0 1 0 0

S1 := 
 0 0 1 0 .
0 0 0 1
By playing the same game with C20 , C30 and
with the matrices

1 0
2 −1
S2 := 
0 0
0 0

1
0
S3 := 
2
0
0
1
2
0
C40 we can make the same assertion

0 0
2 2
,
1 0
0 1
0
0
−1
0

0
0

2
1
22
BEN GREEN
and

1
0
S4 := 
0
2
0
1
0
2
0
0
1
2

0
0
.
0
−1
This leads naturally to consideration of the orbit hS̃ix0 ⊆ Z4 , where
S̃ := {S1 , S2 , S3 , S4 },
every vector in which consists entirely of curvatures of circles in the Apollonian
packing. This puts us in a situation very similar to that studied in Theorem 6.2,
except now we appear to be dealing with a subgroup of GL4 (Z) rather than of
SL2 (Z).
It turns out, however, that this situation is essentially a two-dimensional one in
disguise, and for this we need to add to the list of areas of mathematics we touch
upon by hinting at Lie theory and special relativity! The matrices S1 , S2 , S3 , S4
belong to SOF (Z), the subgroup of GL4 (Z) consisting of 4 × 4 matrices with determinant one which preserve the quadratic form F (~x) = 2(x21 + x22 + x23 + x24 ) − (x1 +
x2 + x3 + x4 )2 (cf. (6.1)). By the standard theory of quadratic forms (over R) this
is equivalent to the Lorentz form L(~y ) = y12 + y22 + y32 − y42 , and so we may identify
SOF (R) with the orthogonal group SO(3, 1) preserving this latter form. But it is
very well-known that this group admits SU(2) as a double cover: this is because
the set {~y : L(~y ) = −1} may be identified with the set of 2 × 2 hermitian matrices
M with determinant 1 via
y4 + y3 y1 − iy2
(y1 , y2 , y3 , y4 ) 7→
,
y1 + iy2 y4 − y3
and so any P ∈ SU(2) gives rise to an element of SO(3, 1) via the transformation
M 7→ P M P ∗ .
By lifting to this double cover the group hS̃i can be lifted to a subgroup of
SL2 (Z[i]). The proof of Theorem 6.2 goes through with relatively minimal changes,
although once again the group generated by S1 , S2 , S3 and S4 contains unipotents
and so, if the aim is simply to find infinitely many circles or pairs/quadruples of
touching circles with almost-prime curvatures, more elementary approaches work
just as well. Those elementary approaches do not, however, give sharp quantitative
results, whereas the techniques we have sketched do. To explain one such result,
imagine Figure 1 being generated as follows. Start with the outer circle (which has
curvature −6) and the three largest inner circles, with curvatures 13, 21 and 24.
This is the first generation. The second generation consists of those circles touching
three from the first generation: they have curvatures 28,37,61 and 124. The third
generation contains those new circles touching three circles from either the first or
the second generations: these have curvatures 45, 60, 69, 93, 124, 132, 133, 156,
220, 292, 301 and 325. Carry on in this vein: the nth generation will contain 4·3n−2
circles.
Theorem 6.3 (Bourgain, Gamburd, Sarnak). The number of circles at generation
n which have prime curvature is bounded by C3n /n, for some absolute constant C.
We conclude by remarking that there are some very interesting unsolved questions connected with Apollonian packings [24]. In that paper the very interesting
APPROXIMATE GROUPS AND THEIR APPLICATIONS
23
question is raised of whether, in Figure 1, a positive proportion of all positive integers appear as curvatures. J. Bourgain has recently indicated to me that he and
Elena Fuchs have obtained new information on this question. See also [39] for an
asymptotic formula for the number of circles in the packing of curvature at most X.
It seems that the question of describing this set of integers more precisely remains
open: are they given, from some point on, by finitely many congruence conditions?
7. Acknowledgements
I am very grateful to Cliff Reiter for his permission to use the image of an
Apollonian packing. In addition to the cited references, the notes from the fourth
Marker Lecture of Terence Tao were very useful in preparing what is written here. I
thank Jean Bourgain, Emmanuel Breuillard, Alex Gamburd, Harald Helfgott, Alex
Kontorovich, Peter Sarnak and Terry Tao for their comments on an earlier draft of
this survey. Finally, I am very grateful to Lilian Matthiesen and Vicky Neale for
correcting a number of typographical errors.
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Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA
Current address: Radcliffe Institute for Advanced Study, 8 Garden Street, Cambridge MA
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